chapter 5 scenario decomposition framework for mixed

Chapter 5Scenario decomposition framework formixed-integer second-stage problem:stochastic Lagrangean decomposition

The most common framework for dealing with uncertainty in optimiz-

ation models is the two-stage stochastic programming. Typically, two-stage

stochastic programming models comprise two types of decisions: first-stage

decision that must be taken prior to knowing the realization of the uncer-

tainty, and second-stage decisions that represent recourse measures that can

be taken after the uncertainty unveils. The objective is then to minimize both

first-stage and expected recourse costs. In some cases, it might also be suit-

able to consider some sort of risk measure together with the expected recourse

cost in order to avoid incurring in high costs for some of the realization of the

uncertainties (You et al., 2009).

If some of the stage-two variables are required to be integer, we have

what is known as a stochastic integer programming (SIP) problem (Carøe and

Schultz, 1999). When this is the case, the stochastic programming problem

loses desirable properties such as convexity and continuity of the recourse

cost function. In this context, solution methods that rely on the use of dual

results from linear programming, such as the L-Shaped algorithm and its

variants (Van Slyke and Wets, 1969; Birge and Louveaux, 1988), cannot be

directly applied to the stochastic problem with second-stage integer variables.

Moreover, the expected recourse function is discontinuous and the set of first-

stage decisions that yield second-stage solutions in known to be nonconvex in

such cases.

In order to deal with this issue, several researchers have proposed ap-

proaches that are capable of dealing with stochastic integer programming prob-

lems. These approaches either try to adapt the L-Shaped algorithm into the

context of stochastic integer programming problems through the use of convexi-

fication techniques for the second-stage problem (see, for example Laporte and

Louveaux (1993); Sherali and Fraticelli (2002); Zheng et al. (2012)), enumer-

ative branch-and-bound strategies (see for example Carøe and Schultz (1999);

DBD

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Chapter 5. Scenario decomposition framework for mixed-integer

second-stage problem: stochastic Lagrangean decomposition 63

Norkin et al. (1998); Ahmed et al. (2004)), or else apply dual decomposi-

tion methods by means of Lagrangean decomposition approaches (Carøe and

Schultz, 1999). In this case, the problem is decomposed into scenario subprob-

lems through the relaxation of non-anticipativity constraints (NAC) and the

solution strategy relies on finding the optimal dual multipliers. Several meth-

ods have been proposed in the literature for solving the dual problem associ-

ated with the Lagrangean decomposition. The techniques available include the

classical subgradient method (Held and Karp, 1971; Held et al., 1974), cutting-

plane approaches (Kelley, 1960; Redondo and Conejo, 1999), the volume al-

gorithm (Barahona and Anbil, 2000), bundle methods (Lemarechal, 1989; Zhao

and Luh, 2002), and augmented Lagrangean methods (Ruszczynski, 1995; Li

and Ierapetritou, 2012).

In this chapter we present a comprehensive framework for the multi-

product, multi-period supply chain investment planning problem considering

network design and discrete capacity expansion under demand uncertainty.

In this case, we consider the existence of integer decisions in the second-stage

problem, which requires a di↵erent approach from the one presented in chapter

4. Some of the novel features that we present include the implementation of a

Lagrangean decomposition to solve a large-scale problem from a real world case

study, an algorithmic approach for solving the dual Lagrangean problem, and a

comparison between the computational performance of di↵erent formulations

for the nonanticipativity conditions (NAC). Furthermore, we consider a risk

measure that allows us to reduce the probability of incurring in high costs

while preserving the decomposable characteristic of the problem.

5.1 Mathematical Model

In chapter 4 we have considered a simplified version of the supply chain

investment planning model presented in chapter 2. In the same spirit of what

was done in chapter 4, we consider here a simplified version of the complete

model 2.1 to 2.18 presented in chapter 2, however with some additional

features.

In this chapter we consider that, since we are dealing with uncertainty

of the demand levels, it might be the case that the base does not have enough

tankage available to deal with unexpectedly high demand. When this is the

case, transportation ships can be used as temporary tanks in places where

marine access is available. This is only done under emergency circumstances

due to the high impact on the logistic costs.

DBD




(a) Nomenclature

The mathematical model considered in this section uses the same nomen-

clature presented in Table 4.1, except for the terms that are not comprised in

the version of the model 4.1 to 4.10 presented in chapter 4. Table 5.1 lists the

additional terms considered.

Parameters

Ejt

Emergency floating tankage cost

Fjt

Emergency floating tankage capacity

Variables

z⇠jt

Emergency tankage contract decision

Table 5.1: Model Additional Notation

(b) Model Formulation

The mathematical model used in this section is very similar to the model

4.1 to 4.10 presented in chapter 4, except for the objective function and

constraints 4.5, 4.8, and 4.10 of the second-stage problem. This is due to the

fact that in this chapter we removed the continuous variable u⇠

jpt

representing

the unmet demand from constraint 4.5 and added the term Fjt

z⇠jt

to constraints

4.8 and 4.10. In addition, we replaced the termP

j,p,t

Sjpt

u⇠

jpt

that represented

the shortfall cost byP

j,t

Ejt

zjt

, which represents the costs for hiring emergency

floating capacity. For the sake of completeness, we state below the second-stage

model considered in this chapter.

Q(w, y) = minx,v2R+

,z2{0,1}

X

⇠

P ⇠

X

i,j,p,t

Cijt

x⇠

ijpt

+X

j,p,t

Hjpt

v⇠jpt

+X

j,t

Ejt

zjt

!

(5.1)

s.t.:X

i

x⇠

ijpt

+ v⇠jpt�1 =

X

i

x⇠

jipt

+ v⇠jpt

+D⇠

jpt

8j 2 B, p 2 P , t 2 T , ⇠ 2 ⌦

(5.2)

v⇠jpt

M0jp

+Mjp

X

t

0t

wjpt

0 + Fjt

z⇠jt

8j 2 B, p 2 P , t 2 T , ⇠ 2 ⌦ (5.3)

X

i

x⇠

ijpt

Kjp

M0

jp

+Mjp

X

t

0t

wjpt

0

!+ F

jt

z⇠jt

8j 2 B, p 2 P , t 2 T , ⇠ 2 ⌦

(5.4)

4.6, 4.7, and 4.9

DBD




The objective function 5.1 represents freight costs between nodes, inventory

costs, and costs for hiring emergency floating capacity. Equation 5.2 comprises

the material balance in distribution bases. Constraint 5.3 defines the storage

capacities together with their expansion possibility and the additional emer-

gency capacity that might be necessary. Constraint 5.4 sets the throughput

limit for bases, defined by the product of the available storage capacity with

the throughput capacity multiplier, and the possibility of expanding them by

means of additional floating tankage.

5.2 Solution Algorithm

From our early investigations, we observed that large-scale instances of

the stochastic supply chain investment planning problem presented in section

5.1 cannot be solved in full space. In this sense, we consider that scenario-

wise Lagrangean decomposition is an alternative to overcome this challenge.

In the following section, we detail the algorithmic strategy for solving the

aforementioned problem. Our method integrates a scenario-wise decomposition

based on Lagrangean decomposition and novel approach for updating the

Lagrangean multiplier set.

(a) Lagrangean Decomposition Approach

To illustrate the proposed approach, we will consider that we have the

supply chain investment planning problem presented in section 5.1 written in

the following compact notation in the reminder of this chapter.

v = minx,y

cx+X

⇠

P ⇠qy⇠ (5.5)

s.t.:

Ax b (5.6)

Tx+Wy⇠ h⇠ 8⇠ 2 ⌦ (5.7)

x 2 {0, 1}n (5.8)

y⇠ 2 Y 8⇠ 2 ⌦ (5.9)

where c is a n-dimensional vector, q is a p-dimensional vector, A is a m ⇥ n

matrix, b is a m-dimensional vector, T and W are matrices of size q ⇥ n and

q⇥p, respectively, and h is q-dimensional vector. In our context, cx represents

our investment costs (i.e., first-stage costs), whileP

⇠

P ⇠qy⇠ represents the

costs with freight, inventory, and emergency tankage acquisition. (i.e., second-

stage costs). The set of constraints Ax b represents constraints 4.2 and 4.3,

DBD




while Tx+Wy⇠ h⇠ represents constraints 4.7, 4.9, 5.3, and 5.4. Finally, set

Y denotes constraints 4.6 and 5.2, as well as restrictions regarding mixed 0-1

variable domains.

As it is commonly known, this class of problem exhibits a block-angular

structure that can be exploited in a decomposition fashion, provided that we

are able to split it into more manageable pieces. One possible way of making it

possible to decompose the problem is to use Lagrangean decomposition (Fisher,

1985; Guignard and Rosenwein, 1989) in the context of stochastic optimization.

Such a procedure was first proposed by (Carøe and Schultz, 1999) allowing the

problem to be decomposed into scenario subproblems. The idea behind this

scenario decomposition approach is to create copies x1, . . . , x|⌦| of the first-

stage variables and then rewrite the problem as follows:

v = minx,y

X

⇠

P ⇠

�cx⇠ + qy⇠

�(5.10)

s.t.:

Ax⇠ b 8⇠ 2 ⌦ (5.11)

Tx⇠ +Wy⇠ h⇠ 8⇠ 2 ⌦ (5.12)

y⇠ 2 Y 8⇠ 2 ⌦ (5.13)

x1 = · · · = x|⌦| (5.14)

Equalities 5.14 correspond to the nonanticipativity constraints (NAC for

short, as defined in the beginning of this chapter). As the name suggests,

these constraints state that the first-stage decisions must not depend on any

particular scenario which will prevail in the second stage. In other words, it

means that we cannot have particular first-stage solutions for specific scenarios

given that these solutions must be taken prior to the uncertainty realization.

There are several ways of representing NAC. They can be expressed

in aggregated form, where a single constraint is used to express the non-

anticipativity property, or considering a disaggregated form, in which indi-

vidual NAC are used to represent the non-anticipativity between the first-stage

variables in a pairwise fashion. In order to be able to decide between these two

representations one must consider the inherent trade-o↵ between them. Even

though the aggregate constraint yields in general weaker linear relaxations

than the conjunction of the NAC, the advantage is that fewer Lagrangean

multipliers are needed, which might make it easier to find good values. Using

the disaggregate formulation requires a larger number of multipliers, although

with the potential benefit of having more control when it comes to the search

of good multiplier values.

DBD




In our early experimentations, we observed that the disaggregated repres-

entation of NAC provided better computational results in our context. There-

fore, we will only consider this type of representation hereinafter. Nevertheless,

even the disaggregated representation can be done in di↵erent manners. In this

thesis, we consider two di↵erent ways of formulating disaggregated NAC.

The first formulation assigns the scenario copy variables in a sequential

fashion. In this way, we can replace condition 5.14 by the following set of

constraints:

x⇠ = x⇠+1 8⇠ = 1, . . . , |⌦|� 1 (5.15)

The other representation consists of associating the scenario copy variables

considering one scenario (say the first scenario) as a reference to other copy

variables. By doing this, an asymmetric structure is created regarding the set

of constraints that represent the non-anticipativity conditions. In this case, the

NAC are formulated as follows:

x1 = x⇠ 8⇠ = 2, . . . , |⌦| (5.16)

Independent of which representation one might choose, the Lagrangean relax-

ation with respect to the non-anticipativity condition 5.14 is the problem of

finding x⇠, y⇠, ⇠ = 1, . . . , |⌦| such that:

D(�) = minx,y

X

⇠

P ⇠

�cx⇠ + qy⇠

�+X

⇠

�⇠s⇠ (5.17)

s.t.:

Ax⇠ b 8⇠ 2 ⌦ (5.18)

T ⇠

⇠

x+W⇠

y⇠ h⇠ 8⇠ 2 ⌦ (5.19)

y⇠ 2 Y (5.20)

where s⇠ = x⇠ � x⇠+1, ⇠ = 1, . . . , |⌦| � 1 for the sequential representation,

s⇠ = x1�x⇠, ⇠ = 2, . . . , |⌦| for the asymmetric representation, and � is (|⌦|�1)-

dimensional vector. The Lagrangean dual then becomes the problem of finding

� such that

vLD

= max�

D(�) (5.21)

Duality theory establishes that v � vLD

Guignard (2003). In particular, for

nonconvex cases such as MILP models, we may have that v > vLD

, which

implies the existence of a duality gap. This fact is a well known result and can

DBD




be found in Carøe and Schultz (1999).

One important property of the Lagrangian dual problem 5.21 is that it

is a convex non-smooth program, which splits into separate subproblems for

each scenario ⇠ that can be solved independently. Each scenario subproblem

can be then stated in the following form:

D⇠(�) = minx,y

P ⇠

�cx⇠ + qy⇠

�+ f ⇠(�)x⇠ (5.22)

s.t.:

Ax⇠ b (5.23)

T ⇠

⇠

x+W⇠

y⇠ h⇠ (5.24)

y⇠ 2 Y (5.25)

where D(�) =P

⇠

D⇠(�) and f ⇠(�) is given depending on the chosen formula-

tion for the NAC. For the sequential case we have that

f ⇠(�) =

8>>><

>>>:

�1, if ⇠ = 1

��|⌦|, if ⇠ = |⌦|

�⇠ � �⇠�1, otherwise

and for the asymmetric formulation, we have that

f ⇠(�) =

8<

:

P|⌦|⇠=2 �

⇠, if ⇠ = 1

��⇠, otherwise

Note that each one of the scenario subproblems are completely independent

and, thus, this type of decomposition could benefit from the use of parallel

computation.

(b) Proposed strategy for solving the Lagrangean Dual

Although computationally convenient, the decomposition framework

presented in section 5.2(a) does not solve the original full-space problem. Nev-

ertheless, it is widely known that the Lagrangean dual represents a relaxation

of the original problem for any given set of Lagrange multipliers (Guignard,

2003). In this sense, we can concentrate our e↵orts in finding better multipliers

sets, i.e. multipliers yielding tighter relaxations to the original problem that

approximate as much as possible the solution of the Lagrangean dual to the

solution of the full-space problem.

Typically, Lagrangean decomposition algorithms rely on successively

DBD




solving the Lagrangean subproblems for sequentially improving Lagrange

multipliers that are obtained based on the use of some information available

after solving these subproblems. The algorithm starts with an initial guess

for the Lagrange multipliers, which can be obtained by some problem-specific

strategy, such as dual values of NAC from the linear relaxation of the complete

problem, or even set to prespecified values (e.g, zero). Then, at each iteration

the Lagrangean subproblems are solved and a relaxed solution is obtained. The

algorithm stops when some of the convergence criteria are satisfied. Otherwise,

the Lagrange multipliers are updated and the algorithm proceeds towards the

next iteration. Figure 5.1 schematically illustrates the proposed algorithm. The

algorithm starts at an initialization step, where the Lagrange multipliers and

other parameters are set to their initial value. The algorithm then proceeds

with the iterative solution of the Lagrangean dual, followed by the procedure

that seeks to obtain a feasible solution from the solution obtained in the

Lagrangean dual. This iterative procedure continues until the Lagrangean dual

solution and the feasible solution are close enough, as measured by some

stoping criteria. If the criteria are not fulfilled yet, the procedure iterates

using information from these two previous steps to update the Lagrangean

multipliers and iterate once more. Notice that, even though the algorithm

follows a classical iterative framework, it has particular features that di↵er

from traditional approaches.

DBD




Figure 5.1: Schematic representation of the proposed Lagrangean decomposi-

tion

The most common method used to obtain solutions to the Lagrangean

dual is the subgradient method (Held and Karp, 1971; Held et al., 1974).

The method relies on the use of subgradient information available after

solving the Lagrangean relaxation to predict improvement directions for the

multipliers, as well as step sizes. Usually, this approach is preferred because

of its ease of implementation added to its capability of predicting reasonably

good Lagrangean multipliers for many cases. Nevertheless, special care must

be taken in terms of selecting good strategies for defining and updating the

subgradient step size.

Unfortunately, it is sometimes reported in the literature that the sub-

gradient approach might fail to achieve regarding its convergence. To circum-

vent this drawback, many researchers have searched for improvements to this

technique over the years.

One alternative considered is the use of cutting-planes for approximating

the Lagrangean dual function. Cutting-plane strategies are available since

the early 60’s after the seminal work of Kelley (1960). Nevertheless, this

approach might require a large number of iterations in order to yield good

approximations of the Lagrangean dual function and thus, good multiplier

DBD




updates. This e↵ect is mainly due to the fact that at the early iterations of

the algorithm, the number of cutting planes generated is too few to provide

good Lagrangean multipliers. One possible way of improving these kind of

approaches is to consider trust-regions for the multipliers in the early iterations,

so that this e↵ect is controlled (Mouret et al., 2011).

In this chapter we introduce a novel hybrid approach where we seek to

combine the ideas from the aforementioned methods into a single framework.

The main idea behind the algorithm is to combine cutting plane generation

using the dual information obtained from the solution of the Lagrangean

dual problem with subgradients that provides approximated ascent directions.

Moreover, we use the step size predicted by the subgradient method as

a trust-region for the multipliers update process. As will be seen in the

computational results section, the combination of these techniques provides

e↵ective Lagrangean multiplier updates, while ensuring good convergence

properties.

(c) Upper bounding procedure

One specific feature of the proposed algorithm is the particular heuristic

that uses information derived from the solution of the Lagrangean dual problem

to derive a feasible solution and a valid upper bound to the full-space problem.

It should be noted that it is not computationally demanding to calculate an

upper bound for the full-space problem, once a first-stage solution is available.

This is mainly due to the fact that for a fixed first-stage solution, the full-space

problem becomes decomposable in scenarios.

The heuristic is based in the following formation rule. Consider a given

iteration K. First, we calculate ↵K

as

↵K

=X

⇠2⌦

P ⇠x⇠

K

�X

⇠2⌦

P ⇠(1� x⇠

K

) (5.26)

If ↵K

> 0, the investment (i.e., a combination of location and product in the

case of capacity expansion or origin and destination, in the case of network

design) is selected to compose the feasible solution. The time period for

the selected investment will be the earliest among the scenarios where the

investment was decided. We choose the earliest time period as the one to

be implemented based on the observation that the costs incurred by recourse

actions are typically larger than the increase in first-stage costs due to investing

earlier in a given project. In addition to that, one might notice that the

existence of more logistic options allows the system to possibly reach more

e�cient and less costly logistics, which yields economies of scale.

DBD




Since we are using a heuristic to generating solutions based on inform-

ation that comes from scenarios individually, it might be the case that the

solution generated is not feasible for the full-space problem. If it is the case,

then we use an integer cut to remove this infeasible solution from the search

space of the relaxed dual. Let X1 = {j | xj

= 1} and X0 = {j | xj

= 0}. Then,we can write the integer cut as

X

j2X0

xj

+X

j2X1

(1� xj

) � 1 (5.27)

and add it to every scenario subproblem. We then solve again the Lagrangean

relaxation and proceed with algorithm execution.

(d) Multiplier updating procedure

The most common method for updating the Lagrangean multipliers is the

subgradient algorithm (Held and Karp, 1971; Held et al., 1974). This algorithm

consists of an iterative method in which at a given iteration K, with a current

set of Lagrangean multipliers �K

a step is taken along the subgradient of D(�).

Let sK

the the subgradient vector of dimension (|⌦| � 1) with components

given as s⇠K

= x⇠

K

� x⇠+1K

, ⇠ = 1, . . . , |⌦| � 1 for the sequential representation

and s⇠K

= x1K

� x⇠

K

, ⇠ = 2, . . . , |⌦| for the asymmetric representation, where

x⇠

K

, ⇠ = 1, ..., |⌦| is the solution for the Lagrangean dual given �K

. Then, the

Lagrange multipliers are updated using the subgradient information as follows:

�⇠

K+1 = �⇠

K

+ ✓K

(UB � LBK

)P

⇠

(s⇠K

)2s⇠K

8⇠ 2 ⌦ (5.28)

where UB is an approximation to the optimal value for v and LBK

= D(�K

).

The term ✓K

2 (0, 2] is used to correct the error in the estimation of the

true optimal value and is usually selected using heuristic rules. The new set

of Lagrangean multipliers �K+1 is then used as an input for solving again the

Lagrangean dual problem. The procedure continues until reaching the limit

in the number of iterations, or unitl some stopping criteria is met, such as

minimum improvement in the magnitude of D(�K

) in some norm (say a l2-

norm) of the subgradient vector (Guignard, 2003).

An alternative procedure for updating the Lagrangean multipliers is

based on the use of cutting planes to approximate the Lagrangean dual func-

tion. In this type of approaches, the solutions obtained from the Lagrangean

dual are used to generate supporting hyperplanes (commonly referred as cuts

in the optimization literature) that iteratively generate an approximation for

DBD




the Lagrangean dual function from which new multipliers are then successively

derived. Given a certain iteration K, the Lagrange multipliers can be obtained

solving the following auxiliary problem:

max⌘,�

⌘ (5.29)

s.t.:

⌘ X

⇠

P ⇠

⇣cx⇠

k

+ qy⇠k

⌘+X

⇠

�⇠s⇠k

8k = 1, . . . , K (5.30)

where 5.30 represents the cuts generated up to iterationK with the information

available in each iteration k = 1, . . . , K. The main drawback associated with

this type of approach is that it is commonly known that this problem is always

unbounded during early iterations of the algorithm (Mouret et al., 2011),

making necessary that some valid bounds are imposed on the multipliers in

order to prevent unboundedness.

In our approach we seek to combine both subgradient and cutting-plan

strategies in a hybrid approach, aiming at developing an e�cient manner

of updating the Lagrangean multiplier set. In the proposed strategy, the

Lagrangean multiplier updates are done by solving the following optimization

problem in a given Kth iteration:

max⌘,�

⌘ (5.31)

s.t.:

⌘ X

⇠

P ⇠

⇣cx⇠

k

+ qy⇠k

⌘+X

⇠

�⇠s⇠k

8k = 1, . . . , K (5.32)

�⇠

K�1 � ✓K

(UB � LBK

)P

⇠

(s⇠K

)2|s⇠

K

| �⇠ �⇠

K�1 + ✓K

(UB � LBK

)P

⇠

(s⇠K

)2|s⇠

K

| 8⇠ 2 ⌦

(5.33)

The objective function 5.31 and constraint 5.32 correspond to the optimiza-

tion problem for the traditional cutting plane approach for updating the Lag-

range multipliers. However, in the proposed algorithm we use a dynamically

updated trust-region for the Lagrangean multipliers in order to circumvent

unboundedness issues. To construct this trust-region, we use subgradient in-

formation available up to the current iteration to define step sizes, in the same

spirit of what is done in the classical subgradient method. However, in our case

the multipliers are selected considering an optimization framework rather than

heuristically updating its values. Constraint 5.33 represents the aforementioned

trust-region.

DBD




Note that we use a step length parameter ✓K

to adjust the length of

the step size. This parameter is dynamically updated during the algorithm

execution following the ideas firstly presented in Barahona and Anbil (2000).

We define three types of iterations according to the dual solution value

D(�K

) obtained in the Kth iteration. The first type is when we observe that

D(�K

) < D(�K�1). This is what is called a ”negative” iteration. In this case we

make ✓K

= ��✓K�1, where 0 < �� < 1. Otherwise, we compute gK

= sK�1 ·sK

and if gK

< 0 it means that a further step in the direction of sK

would have

given a smaller value for D(�K

). In this case we call this iteration ”zero”

and make ✓K

= �0✓K�1, where 0 < �� < �0 < 1. Finally, if gK

� 0, then

this iteration is called ”positive”. In this case we make ✓K

= �+✓K�1, where

�+ > 1.

(e) Algorithm statement

We can summarize the proposed algorithm as follows:

Step 1 : Initialization:

1.1) Set UB = 1; LB = �1; K = 1

1.2) Set initial Lagrangean multiplier �K

values;

Step 2 : Solve Lagrangean dual Problem:

2.1) Solve each independent subproblem 5.17 to 5.20 ;

2.2) Combine subproblem solutions to get the lower bound LBK

=P

⇠

D⇠(�K

);

2.3) If LBK

> LB, then LB = LBK

. Also store solution for generating

cuts later.

Step 3 : Generate first-stage solution and derive UB

3.1) Apply the proposed heuristic for generating a first-stage solution

xK

;

3.2) Obtain v(xK

) evaluating xK

in 5.5-5.9. If xK

is not feasible, add a

integer cut of type 5.27 and return to Step 2

3.3) If v(xK

) < UB, then UB = v(x);

Step 4 : If UB �LB < ✏ or any other criteria, such as time elapsed or number

of iterations are met, stop and return xK

and UB. Otherwise, set K = K + 1

and proceed.

Step 5 Lagrange multiplier update:

5.1) Adjust the step length �;

5.2) Solve 5.31 to 5.33 and retrieve the new set of Lagrangean multipliers

�K

. Return to Step 2 ;

DBD




5.3 Risk Management

In stochastic programming, where uncertain data are modeled as

stochastic processes, the objective function value is a random variable that

can be characterized by a probability distribution. Bearing in mind that the

objective function is given as a combination of the total first-stage cost and

the expected cost of the recourse actions, we actually optimize a function char-

acterizing the distribution of this random variable (i.e., its expected value).

Nevertheless, despite the numerous advantages of representing a random

variable by its expected value, it is important to highlight that this is a risk-

neutral approach. In other words, it means that the remaining parameters

characterizing the distribution associated with random variables are not taken

into consideration by the optimization itself. This might lead to cases where,

even though the expected cost is optimal, the distribution of the objective

function cost might present a significant probability of incurring in higher cost

levels.

To control the risk of expected cost distributions with non-desirable prop-

erties, such as a high probability of incurring in high costs, risk management

constitutes an important issue when formulating stochastic programming mod-

els. The most common way of controlling risk is to include in the model for-

mulation a term that represents the measure of the risk associated with a

profit distribution. Popular risk measures include variance (Markowitz, 1952),

shortfall probability (Browne, 1999), expected shortage (Acerbi et al., 2001),

Value-at-Risk (VaR) (Jorion, 2000), and Conditional Value-at-Risk (CVaR)

(Rockafellar and Uryasev, 2000).

We chose to use the expected shortage as a risk measure. The reasoning

behind this choice is related to the inherent interpretation and computational

complexity of each risk measure. On one hand, using the variance as a risk

measure is not completely adequate in the present context since it penalizes in

the same way scenarios with higher and lower costs, since it only is concerned

with deviations from the expected value. On the other hand, risk measures

such as shortfall probability, VaR, and CVaR has the drawback of increasing

the problem complexity, either by increasing the number of binary variables (in

the case of shortfall probability and VaR), or by destroying the decomposable

structure of the Lagrangian dual (in the case of VaR and CVaR) 2.

The expected shortage can be defined as the expectation of the cost in the

scenarios where the cost is higher than a pre-specified target ⌘. The expected

2One might argue that decomposition can be restored by creating additional copyvariables. Nevertheless, it would increase the size of the multipliers set, and thus, thecomplexity of the problem.

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shortage is given by:

ES(⌘, x) =1

SP (⌘, x)

X

⇠

P ⇠ max{cx+ qy⇠k

� ⌘, 0} (5.34)

where SP (⌘, x) =P

⇠|cx+qy

⇠k>⌘

P ⇠. One can observe that expression max{cx+

qy⇠k

� ⌘, 0} is di↵erent from zero in all scenarios in which the cost is greater

than ⌘. In order to properly calculate the expected value of the cost over

such scenarios, it is necessary not to take into account the probability of those

scenarios with a cost smaller than ⌘. For this reason, the expectation expression

above must be divided by the sum of the probabilities of all scenarios with

a cost larger than ⌘. The sum of these probabilities is called the shortfall

probability.

The expected shortage can be incorporated into the risk-neutral formu-

lation given in section 5.3 as follows

minx,y,r

cx+X

⇠

P ⇠qy⇠ +X

⇠

P ⇠r⇠ (5.35)

s.t.:

Ax b (5.36)

Tx+Wy⇠ h⇠ 8⇠ 2 ⌦ (5.37)

y⇠ 2 Y (5.38)

cx+ qy⇠ � ⌘ r⇠ 8⇠ 2 ⌦ (5.39)

r⇠ � 0 8⇠ 2 ⌦ (5.40)

where r⇠, 8⇠ 2 ⌦ is a continuous and non-negative variable that is equal to

max{cx + qy⇠k

� ⌘, 0}. Once the problem above is solved and the optimal

values for variables r⇠, 8⇠ 2 ⌦ are available, we calculate the expected shortage

ES(⌘, x) as given by

ES(⌘, x) =1P

⇠|r⇠�0 P⇠

X

⇠

P ⇠r⇠ (5.41)

Note that in this case the block-diagonal structure is preserved, which allow us

to use the same ideas presented in section 5.2 as solution strategy. Moreover,

only one constraint and one continuous variable is added to each subproblem,

which means that there is not significant increase in the complexity of the

subproblems. Regarding the selection of target ⌘, there is an inherent trade-

o↵ between the level of risk accepted and how much optimality might be

compromised in order to reach such desired risk protection. In order to

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elucidate this trade-o↵ one might successively solve the problem for di↵erent

target values and then come up with an Pareto optimal frontier considering

the objective function and di↵erent target levels.

5.4 Numerical results

In this section, we present the numerical results for two di↵erent examples

where the proposed framework is applied. All the problems were modeled using

AIMMS 3.11 and solved with CPLEX 12.1 (including the ones within the

decomposition approach) on Intel i7 1.8GHz CPU with 4GB RAM.

(a) Example 1

The first example consists of a small instance, where a simplified version

of the supply chain investment planning problem is considered. The structure

of the network considered can be found illustrated in Figure 5.2. It consists of

five time periods of one year each, one product, three production sites, and 9

demand points, from which 5 are primary bases with marine access (second

row) and four are secondary bases (third row).

Figure 5.2: Network structure of example 1

In this example, only the primary bases can rely on the use of emergency

floating tankage if necessary. We consider twelve options for the network design

(arcs represented with dotted lines in Figure 5.2) and that only primary

bases are able to have investments in tankage expansion. The bases are

organized in five di↵erent regions (represented by gray rectangles) based on

their geographical proximity. Sixteen demand scenarios are considered, where

it is assumed that the demand for each region can either grow or decrease 5%

per year.

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The equivalent deterministic of the two-stage stochastic problem for this

example has 3859 constraints, 2083 continuous variables, 480 binary variables,

and 13045 non-zeros. Tables 5.2 and 5.3 give the optimal first-stage decisions

concerning the selection and timing of both network design and capacity

expansion decision.

BasesPeriods

2 4

B1 X

B2 X

B3 X

B4 X

Table 5.2: Example 1 Capacity expan-

sion decisions

ArcsPeriods

1 2 5

B1-C1 X

B2-C1 X

B2-C2 X

B3-C2 X

B3-C3 X

B4-C3 X

B4-C4 X

B5-C4 X

Table 5.3: Example 1 Network design

decisions

The optimal expected cost is $ 8718.3 million. If we optimize the problem

considering the average values of the random variables, the optimal solution

of this case would be suboptimal for the complete stochastic problem with

a cost of $ 10134.4. The Value of the Stochastic Solution (VSS) (Birge and

Louveaux, 1997), which is given by the absolute di↵erence between the optimal

value of the stochastic program and the objective value calculated using

the solution of the deterministic problem considering average levels for the

stochastic variables, is $ 1416.1 million. The VSS can be seem as a measure of

the savings in cost due to the consideration of uncertainty, indicating in this

case savings of about 16%. The large savings in this case are related to the

high cost of acquiring emergency floating tankage and with the fact that the

project selection when the demand is considered to be its average comprises

fewer projects, making the floating tankage acquisition more often.

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Sequential Formulation Asymmetric Formulation

Subgradient Proposed Subgradient Proposed

Total Time(s) 1800.0 1037.4 394.4 118.0

Iterations 662 316 136 42

UB($ million) 8718.3 8731.7 8722.8 8726.6

LB($ million) 8487.2 8586.2 8584.9 8555.5

% gap 2.65 1.67 1.58 1.96

Table 5.4: Summary of CPU times(s)

Table 5.4 gives a summary of the number of iterations and the com-

putational time required to reach convergence. We compare 4 di↵erent cases

where we combine two di↵erent formulations for the NAC constraints (namely

asymmetric formulation as given in 3.20, and sequential formulation as given

in 3.23) and two di↵erent algorithms (namely the traditional subgradient al-

gorithm, and our proposed hybrid approach). In this example we used a 2%

optimality gap and 1800s as stopping criteria. We consider in this example

�� = 0.8, �0 = 0.99, and �+ = 1.2.

As can be seen in Table 5.4, the asymmetric formulation performs better

in terms of computational time when compared to the sequential formulation,

independently of which solution technique is used (394.4s versus 1800.0s for

the subgradient algorithm and 118.0s versus 1037.4s for our proposed hybrid

approach). In addition to that, our proposed algorithm performs better than

the traditional subgradient algorithm no matter which formulation is used

(1037.4s versus 1800.0s for the sequential formulation and 118.0s versus 394.4s

for the asymmetric formulation). The di↵erences in the bounds obtained are

due to the use of a 2% gap as one of the stop criteria, which invalidates

any comparisons between the bounds obtained with di↵erent combinations

of formulation and solution techniques.

The best combination observed is the use of the asymmetric formulation

combined with the proposed hybrid approach (118.0s). We believe that the

faster performance presented by the asymmetric formulation is related to

the fact that in this formulation the subproblem for ⇠ = 1 combines the

multipliers from all problems, while in the other formulation the multipliers are

considered in a somewhat myopic fashion since only two di↵erent multipliers

are combined in each subproblem. It seems that for this particular case, the

penalties provided by the Lagrangean multipliers tend to be more e↵ective

in the case of the asymmetric formulation since there are fewer iterations,

thus improving convergence. Moreover, when we compare the two di↵erent

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algorithms, one must bear in mind that the subgradients involving only binary

variables provide a very limited set of possibilities (recall that, for each

dimension, it can only assume the values of 1, -1 or 0). Provided this fact,

and that the subgradient is an estimation to the true ascent direction, the

consequences of having poor estimations can be very harmful. Our proposed

approach deals with that issue by using the magnitude of the step as a reference

combined with the outer-approximation of the Lagrangean dual function to

decide the step size update based on an optimization framework. As the

results suggest, this strategy tends to provide better decisions in terms of

the Lagrange multiplier updates. Figures 5.3, 5.4, 5.5, and 5.6 gives the plots

of the convergence profile of each combination, comparing then between the

two algorithms and the two formulations. In these pictures, ”Asymmetric”

represents the use of the asymmetric formulation, ”Sequential” represents the

use of the sequential formulation, ”Proposed” represents our proposed hybrid

algorithm, and ”Subgradient” the traditional subgradient algorithm.

Figure 5.3: Convergence Profile: Subgradient algorithm and sequential formu-

lation

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Figure 5.4: Convergence Profile: Proposed algorithm and sequential formula-

tion

Figure 5.5: Convergence Profile: Subgradient algorithm and asymmetric for-

mulation

Figure 5.6: Convergence Profile: Proposed algorithm and di↵erent algorithms

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(b) Example 2

The second example represents the realistic supply chain investment

planning problem under demand uncertainty described in section 3.3. In this

case we consider four di↵erent products to be distributed between 16 locations

(13 bases, 1 refinery and 2 international suppliers). A total of 28 projects

for tankage expansion and 3 projects for network design were considered

under a planning horizon of 5 years, divided quarterly into 20 periods. All

computational experiments in this instance were solved considering 7200s (2h)

and 2% optimality gap as stopping criteria.

In order to reduce the number of binary variables of the model, we use

a multi-scale definition for the time horizon regarding investment (first-stage)

decisions and planning (second-stage) decisions. In this sense, we aggregate

the investment decisions such that they are considered to be available at the

beginning of each semester (i.e., considering a semiannually divided horizon),

while the planning decisions are taken considering the original quarterly

divided horizon. Such an approach yields an upper bounding approximation to

the original problem. However, in our early experimentations this was shown to

be an acceptably tight approximation, with di↵erences in the objective function

smaller than 0.5% in our case. Figure 5.7 illustrates the di↵erent scales used

for investment decisions and planning decisions.

Figure 5.7: Multi-scale approximation representation

Scenarios Constraints Binary Var. Continuous Var. Non-zeros

25 113822 1890 88283 478780

50 226822 3390 175783 951705

100 452822 6390 350783 1897555

200 904822 12390 700783 3789255

Table 5.5: Deterministic Equivalent Sizes

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ScenariosSequential Formulation Asymmetric Formulation

Full SpaceSubgradient Proposed Subgradient Proposed

25 1203.2 676.5 622.0 482.7 675.5

50 2714.3 1260.9 908.5 507.3 OoT

100 OoT 4625.5 4018.7 1061.5 OoM

200 OoT OoT OoT 6151.8 OoM

Table 5.6: Summary of CPU times(s)

Table 5.5 shows the deterministic equivalent size of the problem con-

sidered for instances of di↵erent sample sizes, while Table 5.6 gives compu-

tational results in terms of solution times. The instances were solved using

the traditional subgradient and the proposed hybrid algorithms considering

the two di↵erent formulation for NAC. We also compare this solution times

with directly solving the full-space deterministic equivalent problem (column

”Full-space”).

As can be seem in Table 5.6, the hybrid approach combined with the

asymmetric formulation of the NAC outperforms the other possible combina-

tions in terms of computational times. Indeed, for the 200 scenario instance,

this is the only algorithm that is able to reach a 2% optimality gap solution

before the time limit of 7,200s. The entries ”OoM” and ”OoT” stands for Out-

of-Memory and Out-of-Time, respectively. The entry Out-of-Memory means

that the available RAM was not su�cient to deal with the deterministic equi-

valent in these cases. The entry Out-of-Time means that the time limit was

reached by the algorithm before the optimality gap limit. We also consider in

this example �� = 0.8, �0 = 0.99, and �+ = 1.2, which are the same values

used in Example 1.

We solve this case study with a sample size of 200 scenarios. Figure 5.8

shows the results in terms of the cost distribution. The objective function of

the stochastic problem is $64283.8 million. The solution of the deterministic

problem considering the average demand levels for the same 200 scenarios is

the suboptimal solution value of $68236.4 million. The VSS for this scenario

sample is thus $3952.6 million, which represents savings of about 5.8%.

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Figure 5.8: Cost distribution for 200 scenarios

In Figure 5.8, the distribution of costs shows that there is a non-negligible

probability of incurring in high costs due to the dispersion that the probability

distribution presents towards its right-hand side. In order to control the risk

of high costs we applied the risk management model presented in section 5.3

for minimizing the expected shortfall. We set the target to $ 68000 million for

the following calculations based on the assumption that we would like to avoid

possible deviations that exceed the expected cost in more than 5% . Figure 5.9

presents the new distribution of costs for the case when risk is incorporated in

the model.

Figure 5.9: Cost distribution for 200 scenarios after risk management

As can be seen in Figure 5.9 the risk management a↵ects the cost

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distribution by reducing the the expected shortfall (i.e., expected cost over the

target), as well as the probability of incurring in higher costs. In this case, the

objective function value increases to $65588.7 million, or 1.3%. The expected

shortfall cost is reduced from $9433 million to $3850.0 million (over the target),

which represents a reduction of 59.3%. Moreover the probability of shortfall

is dropped from 13.5% to 5.5%. Figure 5.10 shows the comparison between

the objective function value distribution before and after the risk management

technique is applied.

Figure 5.10: Cost distribution comparison

5.5 Conclusions

In this chapter, we presented a two-stage mixed-integer linear stochastic

programming approach for the strategic planning of a multi-product, multi-

period supply chain investment planning problem under demand uncertainty.

We developed a comprehensive framework for solving the problem based on

Lagrangean decomposition, exploiting its scenario decomposable structure. In

this context, the use of decomposition presented itself as being imperative, due

to the large size of the full-space problem. We also presented a novel hybrid

algorithmic framework for updating the Lagrangean multiplier set, based on

the combination of cutting-plane, subgradient, and trust-region strategies.

Numerical results suggests that significant savings in computational times can

be achieved by using the proposed strategy.

We also explicitly consider a risk management tool as a mean to reduce

the chances of incurring in high costs. We chose the expected shortfall as a

risk measures, since it presented itself as being more suitable for the presented

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context when compared to several other risk measures. The results suggest that

this risk measure can e�ciently reduce the high cost risks without increasing

the complexity of the problem.

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chapter 5 scenario decomposition framework for mixed

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